Address delegation really used to happen in three sizes: class A, B and C. Class A delegations would be given from a certain address range, class B delegations from a different range etc. Because the different classes used different address ranges you could determine the class by looking at the first part of an address. And this was built into the routing protocols.
- Class A delegations contained 16777216 addresses each
- Class B delegations contained 65536 addresses each
- Class C delegations contained 256 addresses each
This was very inefficient for networks that didn't fit these sizes. A network that needed 4096 addresses would either get sixteen Class C delegations (which would be bad for the global routing table because each of them would have to be routed separately: the class size was built into the protocol) or they would get one Class B delegation (which would waste a lot of addresses).
In 1993 CIDR was introduced. The protocols were adjusted to be able to deal with prefixes of different sizes and it became possible to route (both internally and externally) prefixes like a /30 or a /21 or a /15 etc etc. Anything between /0 and /32 became possible. Organisations that needed 2048 addresses could get a /21: exactly what they would need.
The way you could internally subdivide those addresses was also limited. There were rules on how you could subnet. Originally each subnet within your classful network had to be the same size. You need one subnet with 128 addresses and another subnet with 16 addresses: too bad.
Variable Length Subnet Masking (VLSM) is the internal-network equivalent of CIDR. VLSM has existed longer than CIDR. It was already mentioned in 1985. So CIDR is basically extending VLSM to inter-domain routing. With VLSM your subnets don't all have to be the same size anymore. You can assign a different number of addresses for each subnet, depending on your needs.
These days all routing on the internet is done without classes. A prefix in the routing table might by coincidence (or because of history) match the classful structure, but protocols will no longer assume they can deduce the prefix length (subnet mask) from the first part of the address. All prefix lengths are explicitly communicated: classless.
Saying that an ISP is in charge of a Class C network is similarly obsolete. Addresses are distributed completely classless by the RIRs (Regional Internet Registries, the organisations responsible for delegating addresses to ISPs and businesses with their own independent addresses).
IPv4 addresses classes really don't exist anymore, and have been deprecated in 1993. If you look at old obsolete routing protocols you can of course still see the assumptions they made based on address class, but that was 20 years ago...
Edited to reflect new information in the question:
Based on your multiple edits which finally give the whole picture (I hope), you can try something like this, but using the switch stacking capability in the alternative configuration will give you a better solution.
Your connection to the ISP will need to be on the same subnet as the ISP.
SW1:
track 3 interface GigabitEthernet1/0/1 line-protocol
!
interface GigabitEthernet1/0/1
description #Primary Link WTBB 1#
switchport
switchport access vlan 3
switchport mode access
no shutdown
!
interface GigabitEthernet1/0/2
description #Link to switch 2#
switchport
switchport access vlan 3
switchport mode access
no shutdown
!
interface Vlan3
ip address 10.107.25.11 255.255.255.240
standby 30 ip 10.107.25.10
standby 30 priority 105
standby 30 preempt
standby 30 track 3 decrement 10
no shutdown
!
SW2:
track 3 interface GigabitEthernet1/0/1 line-protocol
!
interface GigabitEthernet1/0/1
description #Primary Link WTBB 2#
switchport
switchport access vlan 3
switchport mode access
no shutdown
!
interface GigabitEthernet1/0/2
description #Link to switch 1#
switchport
switchport access vlan 3
switchport mode access
no shutdown
!
interface Vlan3
ip address 10.107.25.12 255.255.255.240
standby 30 ip 10.107.25.10
standby 30 priority 100
standby 30 preempt
standby 30 track 3 decrement 10
no shutdown
!
This allows the ISP HSRP to work since the ISP routers need to be able to talk to each other on the same VLAN so that the ISP HSRP works.
Alternate (stacked) Configuration:
SW1/2 stack:
interface GigabitEthernet1/0/1
description #Primary Link WTBB 1#
switchport
switchport access vlan 3
switchport mode access
no shutdown
!
interface GigabitEthernet2/0/1
description #Primary Link WTBB 2#
switchport
switchport access vlan 3
switchport mode access
no shutdown
!
interface Vlan3
ip address 10.107.25.10 255.255.255.240
no shutdown
!
This configuration gives you all the benefits of the two separate switches but solves your problem of having two separate switches with separate IP addresses. The stack is configured as a single switch and is managed as a single device. Besides the data sharing, the 3850s can also do power sharing so that if one of them loses power, the other can keep both switches running. You will also save precious public IP addresses.
Best Answer
Most likely if they're a big university they are their own ISP, using BGP to connect their network to the internet via a number of upstream networks.
Nothing stops them from using IP addresses they should not be using, and it would work in their local network. However, it won't work on the Internet. Their upstream networks providing them connectivity should have filters in place which would only allow the university to advertise IP addresses assigned to them. If the direct upstreams wouldn't filter them, the upstreams' upstreams will. And if IP addresses, which are in use by another network, would be used by the university, that other network would become unreachable from the university network.
In addition, there are a number of projects (for example, RIPE RIS and BGPmon) which monitor routing tables and alert on any "illegal" IP advertisement (BGP hijacks and routing anomalies).